whereγ−αa is the supermomentum of the supermultiplet, P =
Xn i=1
λiλ˜i−q , Q+= Xn
i=1
λiη˜+i , Q−= Xn i=1
λiη˜−i −γ− (6.3) and ˜η+a = ¯u+aA η˜A, ˜η−a′ = ¯u−aA ′η˜A. Hence, supermomentum conservation is manifest in addition to momentum conservation, which allows the use of many supersymmetric meth-ods that were developed in the context of amplitudes, such as the supersymmetric form of BCFW recursion relations [77, 78].
The tree-level colour-ordered MHV super form factor of T is [131]2 FˆT,n,2(1, . . . , n;q, γ−) = δ4(P)δ4(Q+)δ4(Q−)
h12ih23i · · · hn−1nihn1i. (6.4) As for amplitudes, we can write the Nk−2MHV form factors of T as
FˆT,n,k(1, . . . , n;q, γ−) = ˆFT,n,2(1, . . . , n;q, γ−)×r(1, . . . , n;ˆ q, γ−), (6.5) where ˆr is of Graßmann degree 4(k−2). Expressions for ˆrofT in momentum-twistor space were given in [131] and [140] at NMHV level and Nk−2MHV level, respectively. Results for certain components can be found in [129, 131].
6.2 On-shell diagrams
In the following section, we introduce on-shell diagrams for form factors.
6.2.1 On-shell diagrams
On-shell diagrams for scattering amplitudes were intensively studied in [74]. They can be used to represent tree-level amplitudes and their unregularised loop-level integrands and furthermore yield the leading singularities of loop-level amplitudes.3 As already mentioned, we will restrict ourselves to tree level. Note that we are using slightly different conventions than [74].
1Following the literature on on-shell diagrams, in this chapter we are suppressing a factor of (2π)4 in each form factor and each amplitude.
2Following the conventions in the literature on form factors ofT, we have absorbed a sign here. Hence, FˆT,2,2(1φ++,2φ++;q, γ−) =−1 in contrast to (2.15).
3For extensions to non-planar amplitudes, see [83–88].
6.2 On-shell diagrams 85 For scattering amplitudes, on-shell diagrams are built from two different building blocks, namely the tree-level three-point MHV and MHV amplitudes, which are depicted as black and white vertices, respectively:4
One way to obtain the on-shell graph for a given scattering amplitude is by constructing the latter via the BCFW recursion relation [77, 78], which can be depicted as [74]5,6
Aˆn,k= X
The representation of an amplitude in terms of BCFW terms, however, is not unique, and neither is the one in terms of on-shell diagrams. Instead, several equivalent represen-tations exist. For planar on-shell diagrams, the set of equivalence relations is generated by two different moves: the merge/unmerge move and the square move [74]. These are depicted in figure 6.1.
(a)Merge/unmerge move for black vertices.
1
Figure 6.1: Equivalence moves between on-shell diagrams for scattering amplitudes. An analogous version of the merge/unmerge move also exists for white vertices.
4In accordance with the literature on on-shell diagrams, we suppress the factor of±i(2π)4 occurring in the amplitudes throughout this chapter.
5Note that we are using BCFW bridges with the opposite assignment of black and white vertices com-pared to [74].
6A more efficient way will be described further below.
For form factors of the stress-tensor supermultiplet, a similar construction via BCFW recursion relations exists [129, 131], which we can depict as:
FˆT,n,k= X
n′,n′′,k′,k′′
n′+n′′=n+2 k′+k′′=k+1
3 n′
n n′+ 1
··· ···
FˆT,n′,k′ Aˆn′′,k′′
2 1
+ 3
n′
n n′+ 1
··· ···
Aˆn′,k′ FˆT,n′′,k′′
2 1
. (6.8)
Via these recursion relations, all ˆFT,n,k can be written in terms of the three-point ampli-tudes (6.6) and the minimal form factor ˆFT,2,2.
In order to introduce on-shell diagrams for form factors, we have to add the minimal form factor as a further building block:7
2 1 =
1 2 = ˆFT,2,2(1,2)
= δ4(λ1λ˜1+λ2˜λ2−q)δ4(λ1η˜+1 +λ2η˜+2)δ4(λ1η˜−1 +λ2η˜−2 −γ−)
h12ih21i .
(6.9) Thus, on-shell diagrams can be used to represent all tree-level form factors — at least for the stress-tensor supermultiplet. It remains to characterise these on-shell diagrams, to find the equivalence relations among them and to find a more direct way to construct them than via BCFW recursion relations.
6.2.2 Inverse soft limits
Another way to construct amplitudes is via the so-called inverse soft limit [80, 241, 242].
This construction amounts to gluing the following structures to two adjacent legs of an on-shell diagram:
, , (6.10)
which respectively preserve the MHV degree k or increase it by one. In principle, all tree-level scattering amplitudes can be constructed via the inverse soft limit [243]. Adding k-preserving and k-increasing structures, however, does not commute. Hence, the inverse soft limit is most powerful for minimal and maximal MHV degree, where only one of the two structures in (6.10) occurs and the order of applying them is irrelevant. Moreover, the inverse soft limit can also be applied to construct tree-level form factors of T [243].
Via the k-preserving inverse soft limit, the four-point MHV amplitude ˆA4,2 can be
7While the three-point amplitudes can be either MHV or MHV, the minimal form factor is both MHV and NmaxMHV. Hence, we only have one form-factor vertex (6.9) as building block, while we have two amplitude vertices (6.6).
6.2 On-shell diagrams 87 constructed from the three-point MHV amplitude ˆA3,2 as
2
1 3
−−−−→
2
3
4
1 , (6.11)
which agrees with the result from the BCFW recursion relation (6.7).8
Similarly, the three-point MHV form factor ˆFT,3,2 can be built from the minimal form factor ˆFT,2,2 as
1 2 −−−−→ 2
3
1 , (6.12)
which agrees with the result from the BCFW recursion relation (6.8).
Note that the on-shell diagram (6.12) for ˆFT,3,2 is not manifestly invariant under cyclic permutations of its three on-shell legs, while ˆFT,3,2 is. Similarly, the on-shell diagram (6.11) for ˆA4,2 is not manifestly invariant under cyclic permutations of its four on-shell legs, although ˆA4,2 is. The equivalence of both cyclic orderings of the on-shell diagram (6.11) is precisely the statement of the square move depicted in figure 6.1b. The cyclic invariance of the on-shell diagrams for all other MHV amplitudes, however, follows from its combination with the merge/unmerge move. Hence, we have to add an additional equivalence move for on-shell diagrams of form factors, which reflects the cyclic invariance of the three-point form factor. This move is depicted in figure 6.2 and we call it rotation move. As the square move for amplitudes, it implies the cyclic invariance of all other MHV form factors when combined with the moves in figure 6.1.
1 2
3 =
2 3
1 =
3 1 2
Figure 6.2: Rotation move for on-shell diagrams that involve the minimal form factor.
In addition to the depicted version with one black and two white vertices, an analogous version with one white and two black vertices exists. Similar to the other moves, this move can be applied to any subdiagram of a given on-shell diagram.
The three-point NMHV form factor ˆFT,3,3 can be built from the minimal form factor FˆT,2,2 by the k-increasing inverse soft limit. The resulting on-shell diagram is related to
8Throughout this chapter, we disregard terms in which external legs are connected by a chain of vertices of the same colour, as these do not contribute for generic external momenta; cf. also the discussion in subsection 3.3.1.
the on-shell diagram (6.12) by inverting the colour of the vertices. As in the MHV case, its cyclic invariance gives rise to another equivalence move: the rotation move with inverted colours.
6.2.3 Permutations
For scattering amplitudes, a permutation σ can be associated with every on-shell diagram [74, 244]. For brevity, we write permutations
σ=
1 2 3 . . . n
↓ ↓ ↓ . . . ↓
σ(1) σ(2) σ(3) . . . σ(n)
(6.13)
as σ = (σ(1), σ(2), σ(3), . . . , σ(n)). The association is as follows. Entering the on-shell diagram at an external legi, turn left at every white vertex and right at every black vertex until arriving at an external leg again, which is then identified as σ(i). For example,
1
3 2
→σ= (3,1,2),
1
3 2
→σ= (2,3,1). (6.14)
Note that the permutation associated with an on-shell diagram is invariant under the equivalence moves in figure 6.1.9
We can define a permutation for on-shell diagrams of form factors by the additional prescription to turn back at the minimal form factor, i.e.
2 1 →σ= (1,2). (6.15)
The resulting permutation is invariant under the equivalence moves in figure 6.1 as well as figure 6.2.
For n-point MHV and MHV scattering amplitudes, the permutations associated with the on-shell diagrams are well known to beσ = (3, . . . , n,1,2) andσ= (n−1, n,1. . . , n−2) [74], respectively. From the construction via inverse soft limits, we find that the permuta-tions associated with the on-shell diagrams of n-point MHV and NmaxMHV form factors are respectively given byσ= (3, . . . , n,1,2) andσ = (n−1, n,1. . . , n−2) as well.
6.2.4 Systematic construction for MHV and NmaxMHV
Using the permutations, the corresponding on-shell diagrams for MHV and MHV am-plitudes can be reconstructed in a systematic way [74]. To this end, the permutation σ is decomposed into a sequence of transpositions of minimal length. Note that mul-tiplication of permutations is understood in the sense of the right action, i.e. σ1σ2 = (σ2(σ1(1)), . . . , σ2(σ1(n))).10 Each transposition (i, j) is associated with a BCFW bridge
9Note that, in contrast to [74], we are not using decorated permutations.
10This is different with respect to [74] and related to the fact that we are using the opposite colour assignment for BCFW bridges.
6.2 On-shell diagrams 89 between legsiandj. Starting from an empty diagram withnvacua,11the on-shell diagram can be built by acting with BCFW bridges, where the order of applying the bridges is the inverse of the order of the multiplication among the transpositions. Then, all vacua are removed from the diagram as well as all lines that are directly connected to them. In the final step, all vertices that have become two-valent by removing these lines are removed while connecting the two lines that enter the vertices. For the three-point MHV amplitude Aˆ3,2, this construction is illustrated in figure 6.3.
σ = (3,1,2) = (2,3)(1,2) −→
− − +
3 2 1
−→
1
3 2
Figure 6.3: Permutation, construction via BCFW bridges and on-shell diagram for ˆA3,2. For MHV and NmaxMHV form factors, we can use an analogous construction. The only difference is that the two left-most vacua or the two right-most vacua have to be replaced by the minimal form factor, respectively. We illustrate this construction for ˆFT,3,2, ˆFT,4,2, FˆT,5,2 and ˆFT,3,3 in figures 6.4, 6.5, 6.6 and 6.7, respectively.
σ = (3,1,2) = (2,3)(1,2) −→
+
3 2 1
−→ 1
2 3
Figure 6.4: Permutation, construction via BCFW bridges and on-shell diagram for ˆFT,3,2.
σ = (3,4,1,2) = (2,3)(3,4)(1,2)(2,3)−→
+ +
4 3 2 1
−→
1 4
3 2
Figure 6.5: Permutation, construction via BCFW bridges and on-shell diagram for ˆFT,4,2.
11These vacua will be given a specific meaning below. For now, they can be understood in a purely symbolic way.
σ= (3,4,5,1,2) = (2,3)(3,4)(4,5)(1,2)(2,3)(3,4) −→
+ + +
5 4 3 2 1
−→
1 5
4
3 2
Figure 6.6: Permutation, construction via BCFW bridges and on-shell diagram for ˆFT,5,2.
σ = (2,3,1) = (1,2)(2,3) −→
−
3 2 1
−→ 1
2 3
Figure 6.7: Permutation, construction via BCFW bridges and on-shell diagram for ˆFT,3,3. 6.2.5 On-shell diagrams at Nk−2MHV and top-cell diagrams
One important difference between MHV, NmaxMHV and general Nk−2MHV is that sums of different BCFW terms occur in the latter case, whereas only one non-vanishing BCFW term contributes in the former cases. As the BCFW terms are represented by on-shell diagrams, this leads to a sum of different on-shell diagrams. For a given amplitude, all these on-shell diagrams can be obtained from a so-called top-cell diagram by deleting edges, which corresponds to taking residues at the level of BCFW bridges.12 The top-cell diagram for amplitudes can be obtained from the permutation
σ = (k+ 1, . . . , n,1, . . . , k). (6.16) For amplitudes, the MHV degreekranges from 2 to n−2. Hence, the simplest amplitude which is neither MHV nor MHV is the NMHV six-point amplitude ˆA6,3.
For form factors of the stress-tensor supermultiplet, the MHV degree k ranges from 2 to n. Hence, the simplest form factor which is neither MHV nor NmaxMHV is the NMHV four-point form factor ˆFT,4,3. We will look at this example in some detail. All BCFW terms with adjacent shifts that contribute to ˆFT,4,3 are shown in figure 6.8. Via the equivalence moves in figures 6.1 and 6.2, the different terms can be shown to satisfy the following identities:
Ai=D(i+2) mod 4, Bi=C(i+2) mod 4. (6.17) Hence, only eight different BCFW terms occur. These BCFW terms can be obtained from the top-cell diagram depicted in figure 6.9 as well as its image under a cyclic shift of
12Note that not all edges are removable. A criterion for removability is given in [74, 244].
6.2 On-shell diagrams 91
Figure 6.8: All BCFW terms of ˆFT,4,3 that arise from adjacent shift. The terms are grouped such that the ith line stems from a shift in the legs iandi+ 1.
the external on-shell legs by two. Note that several differences occur with respect to the amplitude case. While one top-cell diagram suffices to generate all BCFW terms in the case of scattering amplitudes, this is no longer the case for form factors. This can also be seen from the permutation associated with the top-cell diagram of ˆFT,4,3, which is not cyclic.
Moreover, the decomposition of this permutation into transpositions that is required for the construction of the top-cell diagram in terms of BCFW bridges as discussed in the last subsection is not minimal in the sense it is for amplitudes, cf. figure 6.9.13
σ = (4,2,3,1) = (1,2)(3,4)(2,3)(1,2)(3,4)−→
+ −
4 3 2 1
−→
1
2 3
4
Figure 6.9: Permutation, construction via BCFW bridges and top-cell diagram for ˆFT,4,3. For higher n andk, more than one edge has to be deleted to obtain the BCFW terms from the top-cell diagram, which makes an explicit construction of the latter via BCFW terms increasingly tedious. Instead, we employ the following observation. In all cases considered in this chapter, the on-shell diagrams encoding the BCFW terms for the form factor as well as the top-cell diagrams for the form factor can be obtained from their counterparts in the amplitude case with two more legs as follows. We use the moves in figure 6.1 to expose a box at the boundary of the corresponding on-shell diagram in the amplitude case with two more legs and replace this box by the minimal form factor:
n · · · 3 2 1 n+ 2 n+ 1
−→
n · · · 3 2 1
, (6.18)
where the grey area denotes the rest of the on-shell diagram and we have replaced a box at the external legsn+ 1 and n+ 2 for concreteness.
At the level of the BCFW terms, the above relation can be proven as follows. The on-shell diagram for ˆA4,2 is nothing but a box and the on-shell diagram of ˆFT,2,2 can indeed be obtained by replacing this box with the minimal form factor. Recursively constructing Aˆn+2,k via (6.7), boxes can only occur at the boundary of the on-shell diagram. Comparing
13It would be interesting to find a refined version of the construction in the amplitude case that yields the top-cell diagram directly from the permutation.
6.3 R operators and integrability 93